Yes, the Mixolydian should have a flat seven. For me, the best way to represent the modes is to show them using a single root key, say C. Also just use ascending notes from the root. I think you are using different root notes in the diagram.
If you use C as the root, the notes are just C D E F G A B C. The Mixolydian is the V chord in C (a G7), so the notes of the Mixolydian scale would be G A B C D E F G. Play those notes on the bass and you can see that the seventh is flat. As another example in the key of C, the IV Chord (Fmaj7) is Lydian. The scale notes of the Lydian in C are F G A B C D E F. Play that scale on the bass and the shape has a sharp 4 that distinguishes it from the C major scale.
If you use C as the root, the notes are just C D E F G A B C. The Mixolydian is the V chord in C (a G7), so the notes of the Mixolydian scale would be G A B C D E F G. Play those notes on the bass and you can see that the seventh is flat. As another example in the key of C, the IV Chord (Fmaj7) is Lydian. The scale notes of the Lydian in C are F G A B C D E F. Play that scale on the bass and the shape has a sharp 4 that distinguishes it from the C major scale.

